Mathematics for the interested outsider

Cycle Type

One concept I’d like to introduce is that of the “cycle type” of a permutation. This simply counts the number of cycles of various length in a permutation. For example — using the sample permutations from last time — has two cycles of length two, while has one cycle of length three and one “fixed point” — a cycle of length one. No cycle in a permutation in can be longer than , so we only need to count up to . We collect the information in a cycle type into a tuple of the form . The cycle type of is , and that of is .

It should be clear that the sum of the cycle lengths is . In a formula:

That is, the cycle type breaks up or “partitions” into chunks whose total size adds up to . In general, a partition of is a sequence of numbers in nonincreasing order, whose sum is . Thus the cycle type of gives us the partition , while the cycle type of gives us the partition .

One more example, from the beginning: the two-line notation

describing a permutation in has the cycle notation . Its cycle type is , which corresponds to the partition .

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[…] to each symbol in the cycle notation. In particular, any two conjugate permutations have the same cycle type. In fact, the converse is also true: given any two permutations with the same cycle type, we can […]

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This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.